In the Friedmann-Lemaître cosmological models with vanishing cosmological
constant, the curvature is
directly related to the average energy density: the curvature is positive (spherical space) when the density is higher than the
critical value, zero (Euclidean space) if the density is equal to the critical
value, and negative
(hyperbolic space) if density is lower. The curvature thus dictates only the
time evolution: the universe
is (temporally) closed in the spherical case, (temporally) open in the Euclidean and hyperbolic cases.
The Einstein-de Sitter flat model (that some cosmologists estimate favoured
by inflationary models of
the early universe) corresponds to the diagram of the middle. If moreover the simplest topology is
assumed, the curvature dictates also the finite or infinite character of space:
finite in the
spherical case, infinite in the Euclidean and hyperbolic cases. With these two
(unjustified) simplifications, there is strict equivalence between time
finiteness /infiniteness and space
finiteness / infiniteness. In the Friedmann-Lemaître models
with non zero cosmological constant, the curvature is
related to the matter density and the cosmological constant. There is no more direct link
between the curvature and the cosmic dynamics : the universe can be spherical but
temporally open. If, moreover, topology is not the simplest one, there is no more correspondence
between time finiteness / infiniteness and space finiteness / infiniteness.

The darkness of night

If the paradox of the edge made obstacle to the concept of finite space,
the " dark night paradox "
made obstacle to space infinity. The darkness of the night indeed hides
a mystery involving the
cosmos as a whole, its extension and its history. It is stated like
follows: if space is infinite and uniformly
filled with eternal stars, in any direction which one looks at one must end up finding a star on the line of
sight. In other words, the sky background should be a radiant tapestry, continuously made up of
stars, not leaving any place to the dark. Why isn't it thus? The question,
put as of XVIIth century
by Kepler (and later by Olbers), raised tens of explanations and models. The American writer Edgar Poe
provided the first satisfactory answer. In a premonitory text entitled Eureka, Poe explained why
the darkness of the night rested on the finitude of cosmic time. Indeed, as
he pointed out, the light can propagate only at
finite speed. However, in a non-eternal universe, the stars did not always exist. We can thus receive their light only if this one had time to reach us,
i.e. if the stars which emitted it were sufficiently close. Thus, the sky is not uniformly brilliant
because the stars (not necessarily the entire universe) have existed only for a
finite time. By
understanding how the night darkness privided to us a deep teaching about the
time finiteness of
the world, Poe anticipated by several decades the big-bang relativistic models.

The cosmic microwave background radiation

Since the universe has not existed (at least in a state allowing the existence of
stars) for more than a few billion years, the sky background is hardly brilliant. It emits a weak gleam,
unperceivable to our eyes, but that radiotelescopes can collect. Discovered
in 1965, it is the vestige of
dazzling primitive fire cooled by fifteen billion years of time travel.
Whatever space is infinite or not, only a finite volume is accessible to the
observations. The cosmic microwave background radiation
marks a horizon, an ultimate wall against which will stop any observation forever
in the electromagnetic spectrum. Because, in its
early phase during about one million years, the universe did not give anything
to see: the electromagnetic radiation could not propagate, the stars and the galaxies were
not still formed!

The topology of the universe

The questions related to the global shape of space and, in particular, its
finite or infinite
extension, cannot be fully answered by general relativity (a local physical
theory), but by topology
(a global mathematical theory).

Nothing obliges space to have the simplest topology (known as " simply-connected ") because general
relativity does not impose any constraint on the global properties of
space-time. Many topological "alternatives" of three-dimensional spaces
can thus be used to build relevant universe models, i.e. both compatible
with relativity and observations.

Thanks to " multi-connected " topologies, it becomes possible to consider universe models where
space is finite whatever its curvature, even if the matter density and the cosmological constant are
very low.

Historically, W. de Sitter pointed out in 1917 to Einstein that his static and
spherical universe model could put up with a different topology, namely that of projective space. The
difference was not very large because these two alternatives are finite.
The outstanding article by Friedmann, in 1922, makes mention of a
finite Euclidean space form (normally infinite). Einstein remained unaware
of that since, in 1931, he published with de Sitter an article where they
selected the infinite Euclidean universe
model. Only in 1958, Lemaître mentioned the existence of compact hyperbolic spaces,
also suitable for application to big-bang models. In spite of that, the subject
of cosmic topology always
remained confidential and widely ignored by the community of cosmologists.

In addition to the interest of "compactifying" spaces, the multi-connected models
cause many surprises by creating an "illusion of the infinity". Let us see
why. To build multi-connected
spaces, mathematics teach us that one can start from one of the three types
of " ordinary " (simply connected) spaces.
Then, identification between some points change the shape of
space and makes it multi-connected. From this one can build
universe models where
space is finite (although the curvature can be negative or zero) and of
a really small volume. They are
called "small universes". The simplest example is when our space would be a
hypertorus having a radius lower than five billion light-years. In this case,
the light rays would
have had time to turn three times "around" the universe.
That would imply that each cosmic object (each galaxy for example) should appear according to as
many "ghost" images, observable in various areas of the sky. The observed universe thus
appears made up of the repetition of a same set of galaxies, although
viewed at different look-back times.

It is not easy to check if we live or not in a small universe. The ghost images of each "
real " galaxy would appear in different directions, with different
luminosities, under different
orientations, and at different evolutionary times. It would be
practically impossible to recognize them like such! The universe could appear vast to us, "
unfolded ", filled of billion galaxies, while it would actually be much smaller, " folded up " but
containing only a small number of authentic objects. A gigantic cosmic
optical illusion! Of
course, the current observational data make it possible to eliminate the possibility of a too small
universe... If not we would have already recognized, close to us, the multiple
images of our own Galaxy!
Various arguments of this kind, applied to some well-known cosmic objects (e.g. the closest galaxy clusters), make it
possible to exclude a universe whose dimensions would be lower than a few hundreds of million
light-years. However statistical studies on the distribution of galaxy clusters
may reveal in the future the "crumpled" nature of space over a scale of a few billion light-years.

We see a sky filled with galaxies, but its aspect does not make it possible
to decide if the farthest galaxies are not ghost images of closer galaxies. The assumption of a
multi-connected Universe cannot be discarded: the Universe could appear vast
to us, " unfolded ", while it would be actually much smaller and "folded up".

The basics of topology

Topology is the branch of geometry which classifies spaces according to
their global shape. By
definition, spaces belong to a same topological class if they can deduced from each other by continuous deformation,
i.e. without cutting nor tearing. In the case of two dimensional spaces, i.e. surfaces, the sphere, for
example, has the same topology as any ovoid closed surface. But the plane
has a different topology,
since no continuous deformation will give it the shape of a sphere.
For better visualizing what
is topology, start from the ordinary Euclidean plane. It is an infinite 2-dimensional layer (that
one generally imagines as embedded in ordinary 3 dimensional space). Let us
cut out a tape with infinite " length " but
finite width; then let us identify (i.e. restick) the two edges of this
tape: one gets a
cylinder, i.e. a surface whose topology differs from that of the initial
plane. Let us take another
infinite sheet and, this time, cut out it in rectangle.
Let us identify two by two the parallel edges. We obtain a closed (finite) surface.
It is a flat torus.
From a simple paper sheet we could thus define 3 surfaces with different topologies, pertaining to the
same family of locally flat surfaces.
During the XXth century, mathematicians stuck to the classification of three-dimensional spaces.
Like surfaces, 3-spaces can be arranged,
according to the sign of their curvature, into spherical, Euclidean
or hyperbolic types.
Then one counts the topological forms inside each one of these families. There are for
example 18 kinds of three-dimensional spaces with zero curvature. Simplest
is " ordinary "
infinite Euclidean space, the properties of which are teached at school,
but others space forms are closed and finite. It is for example the case of
the hypertorus, which generalizes
in three dimensions the case of the torus.
A hypertorus can be regarded as the interior of an ordinary cube, whose opposite faces
are identified two by two: while leaving by one, one returns immediately by
the opposite. Such a
space is finite. In addition, there are a countable infinity of spaces
forms with positive curvature, all of them
closed, and an infinite number of spaces with negative curve, some closed
(finite), some open
(infinite). To visualize them, one represents them by the interior of a polyhedron of which some
faces are identified two by two.

The five regular polyhedrons, already called upon by Plato for geometrizing the " elements "
Earth, Water, Air, Fire, Quintessence, are used today to represent certain
multi-connected spaces,
on the condition of considering that the faces are identified by pairs according
to specific
geometrical transformations.

A compact hyperbolic space.

The interior of a regular dodecahedron, whose pentagonal faces are identified
("stuck") by pairs,
is a closed space of negative curve. Seen from inside, such a space would
give the impression we live in a
cellular space, paved ad infinitum by dodecahedrons deformed by optical illusions.

Cosmic sets of mirrors

Who wasn't fascinated by sets of mirrors? That it is about the Galerie
des Glaces at the Chateau de Versailles or most modest Palais of the Ices of open attractions, each one is filled with wonder
at the illusion generated by the phantom images. The mirrors conceal certain secrecies of
infinity.

Everyone noted that to put mirrors on the walls of a room gives the illusion
of a larger room.

Let us take a room filled with mirrors on its six walls (floor and ceiling included). If
you penetrate in the room and light some candles, by the play of the multiple reflexions on the walls
you have immediately
the impression to see the infinity, as if you were suspended on the node of a bottomless well,
ready to be swallowed in a direction or another with the least movement.

It could well be thus of cosmic space!

It may be that the topology of the universe is multiconnected, i.e. that space
resembles inside a room papered
with complicated mirrors. This multiconnexity would create additional
paths for the light rays which reach us from the remote galaxies. It would result a great
number of ghost images of these galaxies. The diagrams on the left result
from numerical simulations
of "crumpled" universes, carried out with my collaborators.

top diagram : space is a hypertorus, represented by the interior of a cube
of 5 billion light-years size, whose opposite faces are identical. 50 galaxies are randomly distributed in space.

middle diagram : positions, on a celestial planisphere, of the 50 " original " galaxies.

bottom diagram : appearance of the sky taking account of the multiple
paths of light
rays. Each " real " galaxy generates about fifty "ghost" images.
It is impossible to distinguish the "real" images from the ghost images. If
one points out the
resemblance of this diagram to the appearance of the large scale structure
in the universe, one deduces that it is
quite possible that we live in a cosmic optical illusion, giving the impression
that space is immense, whereas real space is small and "crumpled".

The quantum universe

It is clear that the concept of a small crumpled universe concerns
Parmenidian aesthetics. This one
took besides the step at the majority of modern physicists, who seek to eliminate
infinite quantities from their theories. Space infinity is not the only
infinity occurring in relativistic cosmology. The
theory predicts configurations indeed where certain geometrical
(e.g. curvature) and physical
(e.g. energy density, temperature) quantities become infinite: gravitational singularities.
Most known are the
initial big-bang singularity, and the final singularity hidden at the bottom of a black hole.
The physicists doubt that a theory predicting singularities can be correct. The fact is that
general relativity is incomplete, since it does not take account of the principles of quantum
mechanics. This last governs the evolution of microscopic world, in particular the field of
elementary particles.
Its essential characteristic is to give a " fuzzy " description of the phenomena,
insofar as the events
can be calculated only in terms of probabilities. However, the occurrence of singularities
brings into play the structure of space-time at very small scale. There is
a length (called Planck length, equal to 10^(-33) centimetre) representing the smallest dimension to which space-time can
still be regarded as smooth. Below, even the texture of space-time would not be continuous any
more but, just like the matter and the energy, formed of small grains. The
gravitational infinities would be replaced by quantum fluctuations of space-time.

With " quantum cosmology ", a theory hardly outlined and promised to attractive developments,
are profiled multiple, simultaneous and not-interacting bubble universes, differing from
each other by their geometry, their topology, their fundamental constants of physics.

All these universes would be like the foam of a single Universe, a kind
of infinite and eternal bubbling ocean, in perpetual transformation,
called by the physicists " quantum vacuum ". With such a conception, Heraclitus' sons did not say their last word...

The Foam of Vacuum

Quantum cosmology makes it possible to consider multiple universes, without interaction between
them. Our observable universe would occupy a " bubble " born of the spontaneous fluctuations
of the quantum vacuum, like many other bubbles.

Copyright : Manchu/Ciel et Espace

The black hole infinities

A hybrid creature given birth to by non-Euclidean geometry and relativistic gravitation, the
black hole offers two pretty problems of infinity: a false one and a true
one. A black hole results from
the gravitational collapse of a mass below some critical volume. Like the edge of a bottomless well
dug in the elastic fabric of space-time, its surface - called the event
horizon - marks the geometrical border of a
no return zone. For an external observer, the beats of a clock placed close to the black hole slow
down as the clock is closer to event horizon, until "freezing" when the clock
reaches the surface. All occurs
then as if time were indefinitely delayed. Consequently, the black hole by itself
is inobservable, because it belongs to the infinitely remote future of any observer.
This infinite time is only apparent because it can be
made finite in a correct representation (proper time).
The situation is quite
different with the interior of the black hole.
The general relativity theory predicts the existence of an inescapable singularity inside the black hole, where
the curvature of space and the density of matter become infinite.

A traveller exploring the surroundings of a black hole would be plunged in optical illusions. Misled
by the infinite forgery related to the surface of the hole, it would never see the interior, unless
plunging in person to discover there with its costs the infinite truth of the singularity!

PS : I apologize for my clumsy English. The article is much better written in
French, as you can check here